Try
DSolve[{m x1''[t] + 2 k x1[t] - k x2[t] == 0, m x2''[t] + 2 k x2[t] - k x1[t] == 0, x1'[0] == 0, x1[0] == 0, x2'[0] == 0, x2[0] == L}, {x1, x2}, t]
Try
DSolve[{m x1''[t] + 2 k x1[t] - k x2[t] == 0, m x2''[t] + 2 k x2[t] - k x1[t] == 0, x1'[0] == 0, x1[0] == 0, x2'[0] == 0, x2[0] == L}, {x1, x2}, t]